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09 Jan 2022, 06:35
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Difficulty:
95%
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Question Stats:
22% (02:18) correct
78%
(01:42)
wrong
based on 32
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Can a cube of volume x be fitted inside the cylinder?
(1) Side of the cube and diameter of the base of the cylinder are in ratio 3:4.
(2) If the volume of the cylinder is increased by 50%, 3 cubes, each having its volume equal to x can be fitted inside the cylinder.
(1) Side of the cube and diameter of the base of the cylinder are in ratio 3:4.
(2) If the volume of the cylinder is increased by 50%, 3 cubes, each having its volume equal to x can be fitted inside the cylinder.
09 Jan 2022, 17:17
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AbhiroopGhosh
There are a couple of things that might help us to decide whether the cube can fit in the cylinder. For instance, it would help to know that the volume of the cylinder is greater than x. However, that alone wouldn't help us decide whether the cube would fit; a cylinder could have a bigger volume than the cube but still be incredibly wide and short, for instance, or incredibly narrow and tall. As such, the most important thing to know is whether the dimensions of the cylinder will be big enough to accommodate the dimensions of the cube.
Statement (1) gives us some information about dimensions, so it's promising from the outset, even if it does conspicuously leave out information about the height of the cylinder (to make itself seem insufficient). It can help to diagram this to see if the cube will fit, and because the diameter is the only thing we're given information on (and because determining whether the cube will be able to enter the cylinder at all can be reduced to seeing whether the bottom of the cube, a square, can fit into the top of the cylinder, a circle), it makes sense to model the situation by drawing out a circle and a square. It can also help to make up dimensions for the diameter of the circle and the side of the square (4 and 3, respectively). I've included a diagram below. Note on the diagram that I've included the diagonal of the square. This is key to determining statement (1)'s sufficiency: for the cube to even have a chance at entering the cylinder, the square's biggest dimension (which is always its diagonal) has to be smaller than the diameter of the circle. However, the diagonal of the square here is \(3\sqrt{2}\), and given that \(\sqrt{2}\) is roughly 1.4, the diagonal of the square is roughly 4.2. And because this length will increase in a way that's directly proportional to increases in the sides of the cube and diameter of the circle, we can trust that the cube will always be too big to fit into the cylinder. This gives us a definitive No to the question, and as such, statement (1) is sufficient.
Statement (2) gives us information that indirectly speaks to the volumes of the two shapes. On its surface, it seems as though if an increase of 50% in the volume of the cylinder allows three cubes to fit, the cylinder at its original size would allow 2 cubes to fit. However, this outcome should arouse suspicion: there's no way that statement (2) could be sufficient with an answer of Yes when statement (1) is sufficient with an answer of No (this is always true in DS questions). Furthermore, statement (1) can help by reminding us of what we don't know from this statement: we don't know if the increase of 50% in the volume of the cylinder is such that it changes the cylinder from having a dimension (such as the diameter) that's too small to fit any cubes to having a dimension that's now big enough to accommodate the cubes. We could, for instance, assume that the sides of the cubes and the diameter of the original cylinder were the 3 and 4 (respectively) that they were in statement (1) and that the 50% increase went straight to the diameter of the cylinder, in which case the answer to the question would be a No; similarly, it's easy to come up with an alternative case in which the diameter of the original cylinder was 6, such that the same cube could fit in it, and the 50% increase did just take the capacity of the cylinder from 2 cubes to 3 cubes (for an answer of Yes). Because we can come up with a No case and a Yes case, statement (2) is insufficient.
The answer is (A).
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